Magnetic anisotropy is the directional dependence of a material's magnetic properties. The magnetic moment of magnetically anisotropic materials will tend to align with an "easy axis", which is an energetically favorable direction of spontaneous magnetization. The two opposite directions along an easy axis are usually equivalent, and the actual direction of magnetization can be along either of them (see spontaneous symmetry breaking).
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In contrast, a magnetically isotropic material has no preferential direction for its magnetic moment unless there is an applied magnetic field.
Magnetic anisotropy is a prerequisite for hysteresis in ferromagnets: without it, a ferromagnet is superparamagnetic.
Anisotropy energy of a single-domain magnet
Suppose that a ferromagnet is single-domain in the strictest sense: the magnetization is uniform and rotates in unison. If the magnetic moment is
Uniaxial
A magnetic particle with uniaxial anisotropy has one easy axis. If the easy axis is in the
where
Triaxial
A magnetic particle with triaxial anisotropy still has a single easy axis, but it also has a hard axis (direction of maximum energy) and an intermediate axis (direction associated with a saddle point in the energy). The coordinates can be chosen so the energy has the form
If
Cubic
A magnetic particle with cubic anisotropy has three or four easy axes, depending on the anisotropy parameters. The energy has the form
If